Problem 2 - Understanding Elementary Algebra with Geometry

Question

Complete each ordered pair so that it satisfies the given equation. $$5 y+6 x=30 ; \quad(-5, \quad),(\quad,-6),(\quad, 4)$$

Step-by-Step Solution

Answer

(-5, 12), (10, -6), (\frac{5}{3}, 4)

Step 1: Substitute the first x-value

The given equation is \(5y + 6x = 30\). Substitute \(-5\) for \(x\) in the equation: \[5y + 6(-5) = 30\]

Step 2: Solve for y with the first x-value

Simplify the equation: \[5y - 30 = 30\] Add 30 to both sides: \[5y = 60\] Divide both sides by 5: \[y = 12\] So the first ordered pair is \((-5, 12)\).

Step 3: Substitute the second y-value

Next, substitute \(-6\) for \(y\) in the equation: \[5(-6) + 6x = 30\]

Step 4: Solve for x with the second y-value

Simplify the equation: \[-30 + 6x = 30\] Add 30 to both sides: \[6x = 60\] Divide both sides by 6: \[x = 10\] So the second ordered pair is \((10, -6)\).

Step 5: Substitute the third y-value

Finally, substitute \(4\) for \(y\) in the equation: \[5(4) + 6x = 30\]

Step 6: Solve for x with the third y-value

Simplify the equation: \[20 + 6x = 30\] Subtract 20 from both sides: \[6x = 10\] Divide both sides by 6: \[x = \frac{10}{6} = \frac{5}{3}\] So the third ordered pair is \((\frac{5}{3}, 4)\).

Key Concepts

Ordered Pairs

In this exercise, we work with ordered pairs. An ordered pair consists of two elements, written in a specific order within parentheses, usually as (x, y). These pairs are used to represent points on a coordinate plane. The first element corresponds to the x-coordinate (horizontal position), and the second element corresponds to the y-coordinate (vertical position). For instance, the pair (-5, 12) means you go 5 units to the left and then 12 units up on the grid.Understanding ordered pairs is essential because they provide a simple way to visualize solutions to equations. When working with systems of linear equations, finding the correct ordered pairs that satisfy both equations is often a key goal.

Substitution Method

The substitution method is a powerful algebraic technique used to solve systems of linear equations. This method involves replacing one variable in an equation with an expression derived from another equation. In our problem, we are given the linear equation 5y + 6x = 30 and need to find the y or x value when the other variable is known.

• First, we substitute the given value for x or y into the equation.
• Next, we solve for the remaining variable.

By doing this step-by-step, we isolate and solve for each variable in turn. For example, after substituting -5 for x: 5y + 6(-5) = 30, we solve for y by simplifying and rearranging the equation. This method helps to streamline the process, making it easier to find values that satisfy the equation.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The graph of a linear equation is a straight line. Linear equations can usually be written in the form: ax + by = c, where a, b, and c are constants.In our problem, the linear equation is 5y + 6x = 30. When solving linear equations:

• Identify the given values of x or y.
• Substitute these values into the equation.
• Rearrange the equation to solve for the remaining variable.

This systematic approach ensures accuracy and helps you understand the relationship between variables. By finding ordered pairs that satisfy a linear equation, we can graph these points to visualize the equation.